\section{Simulations}
\label{sec:risk.simulation}

In order to validate our findings, we carried out comprehensive
simulations over a wide range of networks, listed in Table
\ref{tab:networks}. 

\begin{table}[htbp]
\caption{Descriptions of the networks used in the paper. For each network we show its type, name, number of nodes $n$ and edges $m$.}
\label{tab:networks}
\begin{center}
\begin{tabular}{p{1in} p{1.2in} r r p{2.4in}}
\hline
\multicolumn{2}{c}{name} & $n$ & $m$ & description \\
\hline
Human contact & NewRiverValley \cite{barrett:wsc09} & 74,375 & 1,888,833 & Synthetic human contact network for New River Valley county in Virginia. \\
Social communication & Enron mail \cite{enron, enron-link} & 36,691 & 367,666 & Email communication network in a company. \\
Peer-to-peer network & Gnutella \cite{ripeanu+fi:p2p, p2p-link} & 10,876 & 39,994 &Gnutella peer-to-peer file sharing network from August 2002 \\
Random graphs & Preferential attachment \cite{barabasi+perferattach99} & 100,000 & 300,000  & Generated using Python NetworkX library. \\
 & Erd\"{o}s and R\'{e}nyi \cite{erdos1960} & 100,000 & 5,000,000 & \\
\hline
\end{tabular}
\footnotetext[1]{foot note}
\end{center}
\end{table}

The disease transmission is a random process, defined by the parameter
set $\rb{\pt, \pb, \pf, \pv}$. If a node $u$ becomes infectious, it
will infect each of its susceptible neighbors independently with
probability $\pt$, referred as base transmissivity. Each node in the
graph is either vaccinated or not. If a node $u$ is not vaccinated, we
label it as {\uv}. If a node $u$'s vaccine fails, we label it as
{\vf}. Otherwise, we label it {\vs}. Both {\uv} and {\vf} nodes are
susceptible. If a node with vaccine failure is infected then its risk
behavior changes, resulting in boosted transmissivity $\pb$. In the
one-sided model a node infects all its susceptible neighbors with
boosted transmissivity $\pb$, while in the two-sided model it only
infects those neighbors with boosted transmissivity $\pb$ that have
also had a failed vaccination. Parameter $\pv$ denotes the probability
that a node is vaccinated.

In our simulation, every node is labeled with {\uv}, {\vs}, {\vf} with
probability $1-\pv$, $\pv(1-\pf)$, and $\pv\pf$, respectively. All
nodes labeled {\vs} are removed from the graph. Each edge $(u,v)$
connecting two surviving nodes $u$ and $v$ is ``open'', which
corresponds to disease transmission on this edge, or ``close'' with
some probability depending on the model - (i) in the one-sided model,
edge $(u,v)$ is open with probability $\pt$ if both $u$ and $v$ are
labeled {\uv}, and is open with probability $\pb$ if one of $u$ and
$v$ is labeled {\vf}; (ii) in the two-sided model, edge $(u,v)$ is
open with probability $\pt$, unless both $u$ and $v$ are labeled
{\vf}. The closed edges are removed from the graph. In the residual
graph, the connected component containing a specific node $u$ is the
(random) subset of nodes infected, if the disease starts at $u$. Let
$C_1,C_2,\dots,C_k$ be the resulting connected components, then
$\sum_i \abs{C_i}^2/n$ denotes the expected outbreak size of the
disease starting from a random initial node, which we referred as
epidemic severity. Since the disease transmission is a random process,
for a fixed parameter set we run the simulation for 10 iterations, and
take the average value of the epidemic severity. We varified that the
epidemic severity is tightly concentrated around the mean, thus the
average value of the epidemic severity is a good measure.

We want to confirm our findings: (i) both one-sided and two-sided
behavior changes can lead to perverse outcomes (less vaccination is
more effective, more precisely, as the vaccinated fraction $\pv$ is
varied, the epidemic severity shows a non-monotone behavior); (ii) in
both one-sided and two-sided behavior changes, targeted vaccination
can be strictly worse than random vaccination for some level of
vaccine coverage. For each graph, we run simulations over wide range
of parameter set $\rb{\pt, \pb, \pf, \pv}$, and generate the following
4 sets of plots to validate our findings.
%% introduce 4 sets of plots
\begin{itemize}
\item First set of plots shows how the change of boosted
  transmissivity will affect the perverse outcomes, as shown in Figure
  \ref{fig:scalefree.1}, \ref{fig:nrv.1}, \ref{fig:email.1}, and
  \ref{fig:p2p.1}. The $x$-axis is $\pv$ (percentage of vaccinated
  population) and the $y$-axis is the epidemic severity (expected
  percentage of nodes getting infected). We fix the base
  transmissivity $\pt$ and the vaccination success probability $\ps$,
  then plot the curves for different boosted transmissivity.
\item Second set of plots shows how the change of base transmissivity
  will affect the perverse outcomes, as shown in Figure
  \ref{fig:scalefree.2}, \ref{fig:nrv.2}, \ref{fig:email.2}, and
  \ref{fig:p2p.2}. The $x$-axis is $\pv$ and the $y$-axis is the
  epidemic severity. We fix the vaccination success probability $\ps$
  and keep the boosted transmissivity $\pb$ twice the base
  transmissivity $\pt$ (i.e. $\pb=2\pt$), then plot the curves for
  different base transmissivity.
\item Third set of plots shows how the change of vaccination success
  probability will affect the perverse outcomes, as shown in Figure
  \ref{fig:scalefree.3}, \ref{fig:nrv.3}, \ref{fig:email.3}, and
  \ref{fig:p2p.3}. The $x$-axis is $\pv$ and the $y$-axis is the
  epidemic severity. We fix the base transmissivity $\pt$ and the
  boosted transmissivity $\pb$, then plot the curves for different
  vaccination success probability.
\item Fourth set of plots shows the finding that targeted vaccination
  can be strictly worse than random vaccination, as shown in Figure
  \ref{fig:scalefree.4}, \ref{fig:nrv.4}, \ref{fig:email.4}, and
  \ref{fig:p2p.4}. The $x$-axis is $\pv$ and the $y$-axis is the ratio
  between the epidemic severity under targeted vaccination strategy
  and the epidemic severity under random vaccination strategy. If $y$
  value is bigger than $1$, it means targeted strategy is worse than
  random strategy. We fix the base transmissivity $\pt$ and the
  vaccination success probability $\ps$, then plot the curves for
  different boosted transmissivity.
\end{itemize}

In order to capture real disease transmission through simulations, we
find typical values of $R_0$, the basic reproduction number, for many
diseases such as influenza and HIV
\cite{fraser:rzero,chowell:rzero,vynnycky:rzero}. Then, we devide
$R_0$ by the average degree of the graph, and use it as the base
transmissivity $\pt$. For vaccination success probability, we use the
efficacy for real vaccines \cite{karim:science10,gray:hiv,cdc-ve}.

%%%%%%%%%%%%%%%%%%%%%%%
% Scalefree
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/pa_1end_035_025_thick.jpg}
\includegraphics[width=2.5in]{figures/pa_2end_035_025_thick.jpg}
\caption{Epidemic severity with different boosted transmissivities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\pt=0.25$ and $\ps=0.35$.\label{fig:scalefree.1}}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/pa_1end_ps035_thick.jpg}
\includegraphics[width=2.5in]{figures/pa_2end_ps035_thick.jpg}
\caption{Epidemic severity with different transmissivities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\ps=0.35$, and $\pb=2\pt$.\label{fig:scalefree.2}}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/pa_1end_pb05_thick.jpg}
\includegraphics[width=2.5in]{figures/pa_2end_pb05_thick.jpg}
\caption{Epidemic severity with different intervention success probabilities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\pt=0.25$, and $\pb=0.5$.\label{fig:scalefree.3}}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/pa_cmp_1end_035_025_thick.jpg}
\includegraphics[width=2.5in]{figures/pa_cmp_2end_035_025_thick.jpg}
\caption{Epidemic severity comparison of random and targeted intervention strategies in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the ratio of the epidemic severity in targeted intervention strategy and the epidemic severity in random intervention strategy. $\pt=0.25$, and $\ps=0.35$.\label{fig:scalefree.4}}
\end{center}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
% New River Valley
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/nrv_1end_035_003_thick.jpg}
\includegraphics[width=2.5in]{figures/nrv_2end_035_003_thick.jpg}
\caption{Epidemic severity with different boosted transmissivities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\pt=0.03$ and $\ps=0.35$.\label{fig:nrv.1}}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/nrv_1end_035_varyp_thick.jpg}
\includegraphics[width=2.5in]{figures/nrv_2end_035_varyp_thick.jpg}
\caption{Epidemic severity with different transmissivities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\ps=0.35$, and $\pb=2\pt$.\label{fig:nrv.2}}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/nrv_1end_003_varyps_thick.jpg}
\includegraphics[width=2.5in]{figures/nrv_2end_003_varyps_thick.jpg}
\caption{Epidemic severity with different intervention success probabilities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\pt=0.03$, and $\pb=0.06$.\label{fig:nrv.3}}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/nrv_cmp_1end_035_003_thick.jpg}
\includegraphics[width=2.5in]{figures/nrv_cmp_2end_035_003_thick.jpg}
\caption{Epidemic severity comparison of random and targeted intervention strategies in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the ratio of the epidemic severity in targeted intervention strategy and the epidemic severity in random intervention strategy. $\pt=0.03$, and $\ps=0.35$.\label{fig:nrv.4}}
\end{center}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Email
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/email_1end_035_015_thick.jpg}
\includegraphics[width=2.5in]{figures/email_2end_035_015_thick.jpg}
\caption{Epidemic severity with different boosted transmissivities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\pt=0.15$ and $\ps=0.35$.\label{fig:email.1}}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/email_1end_035_varyp_thick.jpg}
\includegraphics[width=2.5in]{figures/email_2end_035_varyp_thick.jpg}
\caption{Epidemic severity with different transmissivities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\ps=0.35$, and $\pb=2\pt$.\label{fig:email.2}}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/email_1end_015_varyps_thick.jpg}
\includegraphics[width=2.5in]{figures/email_2end_015_varyps_thick.jpg}
\caption{Epidemic severity with different intervention success probabilities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\pt=0.15$, and $\pb=0.3$.\label{fig:email.3}}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/email_cmp_1end_035_015_thick.jpg}
\includegraphics[width=2.5in]{figures/email_cmp_2end_035_015_thick.jpg}
\caption{Epidemic severity comparison of random and targeted intervention strategies in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the ratio of the epidemic severity in targeted intervention strategy and the epidemic severity in random intervention strategy. $\pt=0.15$, and $\ps=0.35$.\label{fig:email.4}}
\end{center}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%
% P2P
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/p2p_1end_035_02_thick.jpg}
\includegraphics[width=2.5in]{figures/p2p_2end_035_02_thick.jpg}
\caption{Epidemic severity with different boosted transmissivities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\pt=0.2$ and $\ps=0.35$.\label{fig:p2p.1}}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/p2p_1end_035_varyp_thick.jpg}
\includegraphics[width=2.5in]{figures/p2p_2end_035_varyp_thick.jpg}
\caption{Epidemic severity with different transmissivities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\ps=0.35$, and $\pb=2\pt$.\label{fig:p2p.2}}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/p2p_1end_02_varyps_thick.jpg}
\includegraphics[width=2.5in]{figures/p2p_2end_02_varyps_thick.jpg}
\caption{Epidemic severity with different intervention success probabilities in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the expected percentage of nodes getting infected. $\pt=0.2$, and $\pb=0.4$.\label{fig:p2p.3}}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=2.5in]{figures/p2p_cmp_1end_035_02_thick.jpg}
\includegraphics[width=2.5in]{figures/p2p_cmp_2end_035_02_thick.jpg}
\caption{Epidemic severity comparison of random and targeted intervention strategies in one-sided (left) and two-sided (right) risk behavior models. $x$-axis is the percentage of nodes taking interventions, and $y$-axis is the ratio of the epidemic severity in targeted intervention strategy and the epidemic severity in random intervention strategy. $\pt=0.2$, and $\ps=0.35$.\label{fig:p2p.4}}
\end{center}
\end{figure}

\junk{
%%%%%%%%%%%%%%%% begin junk %%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{figures/pa_1end_08_01.jpg}
\includegraphics[width=3in]{figures/pa_2end_08_01.jpg}
\includegraphics[width=3in]{figures/pa_deg_1end_08_01.jpg}
\includegraphics[width=3in]{figures/pa_deg_2end_08_01.jpg}
\includegraphics[width=3in]{figures/pa_cmp_1end_08_01.jpg}
\includegraphics[width=3in]{figures/pa_cmp_2end_08_01.jpg}
\caption {Variation in the epidemic severity with $\pv$ for the
  one-sided and two-sided models of behavior changes in preferential
  attachment graphs. $\pt=0.1$ and $\pf=0.2$. $x$-axis is $\pv$ and
  $y$-axis is epidemic severity. On the top left are curves for
  different values of boosted transmissivity $\pb$ in the one-sided
  model with random vaccination. On the top right are the curves for
  different values of boosted transmissivity $\pb$ in the two-sided
  model with random vaccination. On the middle left are curves for
  different values of boosted transmissivity $\pb$ in the one-sided
  model with targeted vaccination. On the middle right are curves for
  different values of boosted transmissivity $\pb$ in the two-sided
  model with targeted vaccination. On the bottom left are curves for
  comparison between targeted and randomized vaccination strategies in
  the one-sided model. On the bottom right are curves for comparison
  between targeted and randomized vaccination strategies in the
  two-sided model. $y$-axis here is the ratio between targeted and
  randomized epidemic severity.}
\label{fig:scalefree0801}
\end{center}
\end{figure}


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{figures/pa_1end_06_02.jpg}
\includegraphics[width=3in]{figures/pa_2end_06_02.jpg}
\includegraphics[width=3in]{figures/pa_deg_1end_06_02.jpg}
\includegraphics[width=3in]{figures/pa_deg_2end_06_02.jpg}
\includegraphics[width=3in]{figures/pa_cmp_1end_06_02.jpg}
\includegraphics[width=3in]{figures/pa_cmp_2end_06_02.jpg}
\caption {Variation in the epidemic severity with $\pv$ for the
  one-sided and two-sided models of behavior changes in preferential
  attachment graphs. $\pt=0.2$ and $\pf=0.4$. $x$-axis is $\pv$ and
  $y$-axis is epidemic severity. On the top left are curves for
  different values of boosted transmissivity $\pb$ in the one-sided
  model with random vaccination. On the top right are the curves for
  different values of boosted transmissivity $\pb$ in the two-sided
  model with random vaccination. On the middle left are curves for
  different values of boosted transmissivity $\pb$ in the one-sided
  model with targeted vaccination. On the middle right are curves for
  different values of boosted transmissivity $\pb$ in the two-sided
  model with targeted vaccination. On the bottom left are curves for
  comparison between targeted and randomized vaccination strategies in
  the one-sided model. On the bottom right are curves for comparison
  between targeted and randomized vaccination strategies in the
  two-sided model. $y$-axis here is the ratio between targeted and
  randomized epidemic severity.}
\label{fig:scalefree0602}
\end{center}
\end{figure}


\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{figures/email_1end_08_005.jpg}
\includegraphics[width=3in]{figures/email_2end_08_005.jpg}
\includegraphics[width=3in]{figures/email_deg_1end_08_005.jpg}
\includegraphics[width=3in]{figures/email_deg_2end_08_005.jpg}
\includegraphics[width=3in]{figures/email_cmp_1end_08_005.jpg}
\includegraphics[width=3in]{figures/email_cmp_2end_08_005.jpg}
\caption{Variation in the epidemic severity with $\pv$ for the
  one-sided and two-sided models of behavior changes in Enron email
  network. $\pt=0.1$ and $\pf=0.2$. $x$-axis is $\pv$ and $y$-axis is
  epidemic severity. On the top left are curves for different values
  of boosted transmissivity $\pb$ in one-sided model with random
  vaccination. On the top right are the curves for different values of
  boosted transmissivity $\pb$ in two-sided model with random
  vaccination. On the middle left are curves for different values of
  boosted transmissivity $\pb$ in the one-sided model with targeted
  vaccination. On the middle right are curves for different values of
  boosted transmissivity $\pb$ in the two-sided model with targeted
  vaccination. On the bottom left are curves for comparison between
  targeted and randomized vaccination strategies in the one-sided
  model. On the bottom right are curves for comparison between
  targeted and randomized vaccination strategies in the two-sided
  model. $y$-axis here is the ratio between targeted and randomized
  epidemic severity.}
\label{fig:email}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{figures/p2p_1end_08_01.jpg}
\includegraphics[width=3in]{figures/p2p_2end_08_01.jpg}
\includegraphics[width=3in]{figures/p2p_deg_1end_08_01.jpg}
\includegraphics[width=3in]{figures/p2p_deg_2end_08_01.jpg}
\includegraphics[width=3in]{figures/p2p_cmp_1end_08_01.jpg}
\includegraphics[width=3in]{figures/p2p_cmp_2end_08_01.jpg}
\caption{ Variation in the epidemic severity with $\pv$ for the
  one-sided and two-sided models of behavior changes in Gnutella
  peer-to-peer network. $\pt=0.1$ and $\pf=0.2$. $x$-axis is $\pv$ and
  $y$-axis is epidemic severity. On the top left are curves for
  different values of boosted transmissivity $\pb$ in one-sided model
  with random vaccination. On the top right are the curves for
  different values of boosted transmissivity $\pb$ in two-sided model
  with random vaccination. On the middle left are curves for different
  values of boosted transmissivity $\pb$ in one-sided model with
  targeted vaccination. On the middle right are curves for different
  values of boosted transmissivity $\pb$ in two-sided model with
  targeted vaccination. On the bottom left are curves for comparison
  between targeted and randomized vaccination strategies in one-sided
  model. On the bottom right are curves for comparison between
  targeted and randomized vaccination strategies in two-sided
  model. $y$-axis here is the ratio between targeted and randomized
  epidemic severity.}
\label{fig:p2p}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3in]{figures/nrv_1end_08_002.jpg}
\includegraphics[width=3in]{figures/nrv_2end_08_002.jpg}
\includegraphics[width=3in]{figures/nrv_deg_1end_08_002.jpg}
\includegraphics[width=3in]{figures/nrv_deg_2end_08_002.jpg}
\includegraphics[width=3in]{figures/nrv_cmp_1end_08_002.jpg}
\includegraphics[width=3in]{figures/nrv_cmp_2end_08_002.jpg}
\caption{ Variation in the epidemic severity with $\pv$ for the
  one-sided and two-sided models of behavior changes in New River
  Valley human contact network. $\pt = 0.02$ and $\pf = 0.2$. $x$-axis
  is $\pv$ and $y$-axis is epidemic severity. On the top left are
  curves for different values of boosted transmissivity $\pb$ in
  one-sided model with random vaccination. On the top right are the
  curves for different values of boosted transmissivity $\pb$ in
  two-sided model with random vaccination. On the middle left are
  curves for different values of boosted transmissivity $\pb$ in
  one-sided model with targeted vaccination. Onthe middle right are
  curves for different values of boosted transmissivity $\pb$ in
  two-sided model with targeted vaccination. On the bottom left are
  curves for comparison between targeted and randomized vaccination
  strategies in one-sided model. On the bottom right are curves for
  comparison between targeted and randomized vaccination strategies in
  two-sided model. $y$-axis here is the ratio between targeted and
  randomized epidemic severity.}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%% end junk %%%%%%%%%%%%%%%%%%%%%%
}
